Mixed divided-Frobenius jets and full rank near the dimension count

Mira Cairn

1. Results and relation to earlier work

A polynomial jet matrix over an imaginary quadratic order can lose most of its rank when reduced modulo an inert prime. We retain successive divided Frobenius layers before reducing, and normalize the higher Hasse columns so that the retained information remains integral. A coefficient determinant for an explicit differential connection then proves full row rank. The criterion permits derivative orders larger than the auxiliary prime and approaches the necessary column count.

Let $\mathcal O$ be the maximal order of an imaginary quadratic field and write $\overline z$ for conjugation. For positive integers $K,S$, put $R=KS$ and

$$ \Lambda=\{(r,s)\in\mathbb Z_{\ge0}^2:r+Ks<R\},\qquad M=|\Lambda|=\frac{KS(S+1)}2. $$

At integral nodes $\lambda_1,\ldots,\lambda_n$, the matrix in question is

$$ A_{(r,s),(i,j)} =[t^j](\lambda_i+t)^r\overline\lambda_i^{\,s}, \qquad (r,s)\in\Lambda,\quad 0\le j<J. \tag{1.1} $$

Its entries are Hasse derivatives, with the conjugate variable held fixed. Necessarily $nJ\ge M$ for full row rank.

Theorem 1 (mixed divided-Frobenius rank). Suppose that $p$ is inert in $\mathcal O$, the nodes have distinct residues modulo $p$, and

$$ K<p<R\le p^2,\qquad 1\le J\le R. $$

Put $d=\lfloor(R-1)/p\rfloor<p$ and, for $0\le m\le d$, set

$$ \begin{aligned} D_m&=p\left\lfloor\frac{R-mp-1}{K}\right\rfloor +((R-mp-1)\bmod K),\\ L_\ell&=\min(p,\max(0,J-\ell p)),\\ \mu_e&=\sum_{\ell=0}^e\max(0,L_\ell-e+\ell),\\ B_e&=\sum_{m=0}^eD_m-\frac{e(e+1)}2. \end{aligned} \tag{1.2} $$

If $n\mu_e>B_e$ for every $0\le e\le d$, then (1.1) has full row rank over the quadratic field.

No generic-position assumption is used. In particular, $J\le p$ is not required. A simpler sufficient condition, proved in Section 3, is

$$ n\ge2S,\qquad n\left[J-\frac{RJ}{p^2}-\frac Rp\right]>\frac{(R+p)^2}{2K}. \tag{1.3} $$

With fixed $K$, $J/R\to\epsilon\in(0,1/4)$, $p/R\to0$ and $R/p^2\to0$, its least prescribed node count satisfies $nJ/M\to1$, subject to the inertness and distinct-residue hypotheses.

Theorem 2 (an explicit quadratic-node family). For every negative fundamental discriminant $-D$ with $h(-D)=1$ and $D\ge10000$, define

$$ \begin{aligned} K&=43,& S&=\left\lfloor\frac{29(D-3)}{172}\right\rfloor,& R&=43S,\\ J&=\left\lfloor\frac{13R}{200}\right\rfloor,& A_0&=\left\lceil\frac{11}{10}D^{3/4}\right\rceil,& H&=J-\frac{RJ}{A_0^2}-\frac R{A_0},\\ n&=\max\left(2S,1+\left\lfloor\frac{(R+2A_0)^2}{86H}\right\rfloor\right). \end{aligned} \tag{1.4} $$

Then $H>0$, $n\le\lceil3D/2\rceil$, and (1.1) has full row rank at the $n$ nearest quadratic integers to the origin. The parameter rule has $nJ/M=1+O(D^{-1/4})$. This asymptotic concerns the explicit rule, not the existence of infinitely many class-number-one fields.

Theorem 3 (complete discriminant application). The rank theorem, published local determinant and disk-Sobolev lemmas, and the directed certificates described in Sections 5–7 prove

$$ h(-D)=1\quad\Longrightarrow\quad D<9500. \tag{1.5} $$

Thus one tenth-discriminant bound is $B=9499$, with $\log B=9.158941808890182446528155948973\ldots$. The proof uses neither a global discriminant bound nor a census, known classification or finite elimination. Its local inputs remain explicitly imported. The endpoint is weaker than the reviewed $D\le499$ in Severi Jet [1]; the contribution is the rank mechanism and its complete application. Section 8.1 additionally compares the new and published node rules at the same large support. Its positive comparison at $D=499$, combined separately with [1]’s global bound, gives the qualified corollary $D\le498$.

The local divisibility tower and nearest-node covering estimate originate in Severi Jet's framework [1]. Earlier lower-support Wronskians [2] prove independence by a coefficient determinant in characteristic zero, with an overhead depending on the conjugate degree. The complementary-rank argument [3] combines independent subspaces when full ambient rank is unavailable. Our theorem instead proves the entire triangular row space independent. The complete disk-Sobolev estimates and exact degree counting are imported from [4], using the classical disk-polynomial identities of Figueroa [5]. We do not claim new metric inequalities here.

The inert Frobenius congruence and its use in determinant nonvanishing are classical. Van Dijk [6] uses an auxiliary inert prime and a polynomial root count; twinheterodox [7, Section 5] uses the confluent version in explicit discriminant comparisons. Our additional step retains the divided layers of the evaluated matrix and the higher derivative columns, producing a mixed local interpolation problem. Positive-characteristic connections, $p$-curvature and cyclicity are classical as well; Bostan–Caruso–Schost [8] provide relevant background. We give the elementary connection argument in full. Bertrand's postulate [9] supplies the uniform auxiliary prime. None of these classical tools is claimed as new.

The general proof and exact finite controls were developed by the author. The controls use separately assembled algebraic expressions but are not independent implementations by another researcher. The claimed numerical inequalities are directed certificates, not floating optimization results.

2. Proof of the mixed rank theorem

2.1 Integral triangular layers

Put $g=z^p-w$. The polynomials

$$ g^m z^a w^s,\qquad 0\le a<p,\quad a+Ks<R-mp,\quad0\le m\le d \tag{2.1} $$

form an integral unitriangular basis of the space spanned by $\Lambda$. Indeed, the leading monomial is $z^{mp+a}w^s$, and every replacement $z^p\mapsto-w$ lowers the weight by $p-K>0$. Each original monomial occurs exactly once as a leading monomial, by Euclidean division of its $z$ degree by $p$.

Since $p$ is inert, $\mathcal O/p\mathcal O=k=\mathbb F_{p^2}$ and $\overline\lambda\equiv\lambda^p\pmod p$. At a fixed node write

$$ g(\lambda+t,\overline\lambda)=pU(t)+t^p,\qquad\deg U<p, \tag{2.2} $$

where $U$ has coefficients in $\mathcal O$. Let $\xi$ be the residue of $\lambda$, and let $u(t)=U(t)\bmod p$. Its constant is the residue of $(\lambda^p-\overline\lambda)/p$; it need not be zero.

Conjugation is a Frobenius lift at the inert prime. Thus this constant is the negative of the associated Fermat quotient, or $p$-derivation, in the standard terminology of [10, Section 3]. The arithmetic-jet interpretation is classical; the matrix scaling and rank criterion are the use made here.

For column order $j=qp+v$, $0\le v<p$, multiply that column by $p^q$. Divide each evaluated row in layer $m$ by $p^m$. The resulting entries remain integral: in the expansion of $(pU+t^p)^m$, a term contributing to order $j$ uses $q'\le q$ copies of $t^p$ and has factor $p^{m-q'}$. After scaling it has factor $p^{q-q'}$. Modulo $p$, only $q'=q$ survives, giving

$$ \binom mq[t^v]u(t)^{m-q}(\xi+t)^a\xi^{ps}, \tag{2.3} $$

with value zero for $q>m$. All scalings are invertible in characteristic zero. They will be used only for rank, not as changes to the later metric or determinant valuation normalization.

2.2 A row relation gives mixed local jets

Suppose the scaled matrix has a nonzero row relation over $k$. Collect the coefficients in layer $m$ into

$$ H_m(z)=\sum_{a,s}c_{m,a,s}z^{a+ps},\qquad F(z,u)=\sum_{m=0}^dH_m(z)u^m. \tag{2.4} $$

The exponents $a+ps$ in each layer are distinct because $0\le a<p$. Hence $F\ne0$. Maximizing $a+ps$ subject to $a+Ks\le R-mp-1$ gives $\deg H_m\le D_m$: since $p>K$, the maximum has $s=\lfloor(R-mp-1)/K\rfloor$ and $a=(R-mp-1)\bmod K$.

Write $u_i(z)$ for the reduced local polynomial expressed in the coordinate $z-\xi_i$ at node $i$. Formula (2.3) is equivalent to

$$ (\partial_u^{[\ell]}F)(z,u_i(z)) =O((z-\xi_i)^{L_\ell}), \qquad L_\ell>0. \tag{2.5} $$

Here $\partial_u^{[\ell]}$ denotes the Hasse derivative. The replacement of the fixed factor $\xi_i^{ps}$ by $z^{ps}$ is legitimate because $(\xi_i+t)^{ps}\equiv\xi_i^{ps}\pmod{t^p}$ and every $L_\ell\le p$.

Differentiating (2.2) in characteristic zero and reducing gives the exact local identity

$$ u_i'(z)=z^{p-1}-(z-\xi_i)^{p-1}. \tag{2.6} $$

Thus the error in replacing the derivative along the local curve by the connection below begins at order $p-1$.

2.3 The cyclic differential connection

Set $E=k(z)$ and

$$ \Delta=\partial_z+z^{p-1}\partial_u \quad\text{on }E[u]. $$

Its $p$th iterate is a derivation, since all intermediate binomial coefficients in the iterated Leibniz rule vanish. It sends $z$ to zero, sends $u$ to $(p-1)!=-1$, and kills $k$. Consequently

$$ \Delta^p=-\partial_u. \tag{2.7} $$

Let $e=\deg_uF\le d<p$. We claim that $F,\Delta F,\ldots,\Delta^eF$ are linearly independent over $E$. If the first dependence occurs at index $r\le e$, the span $V$ of the first $r$ elements is stable under $\Delta$. The Leibniz rule accounts for the derivatives of coefficients in $E$. Equation (2.7) makes $V$ stable under $\partial_u$. It therefore contains $F,\partial_uF,\ldots,\partial_u^eF$, which have distinct degrees and nonzero leading coefficients because $e<p$. This forces $\dim_E V\ge e+1$, a contradiction.

The polynomial coefficient determinant

$$ W_F(z)=\det\big([u^m]\Delta^jF\big)_{0\le j,m\le e} \tag{2.8} $$

is therefore nonzero. The degree restriction $e<p$ is essential to this argument. For example, $F=u^p$ is nonconstant in $u$ but $\Delta F=0$. We do not extend Theorem 1 beyond $R\le p^2$.

2.4 Degree and local multiplicity

The degree caps satisfy $D_m-D_{m+1}\ge p$. For an explicit check, write $p=Kq+b$ and $R-mp-1=Kc+r$, with $0\le b,r<K$. If $r\ge b$ the drop is $pq+b$; otherwise it is $p(q+1)+b-K$. Both are at least $p$ because $p>K$. The recurrence

$$ [u^m]\Delta F=H_m'+(m+1)z^{p-1}H_{m+1} $$

then gives $\deg([u^m]\Delta^jF)\le D_m-j$ by induction. Thus

$$ \deg W_F\le\sum_{m=0}^eD_m-\frac{e(e+1)}2=B_e. \tag{2.9} $$

At node $i$, translate the coefficient matrix in (2.8) by $u\mapsto u+u_i(z)$. This has determinant one. Its row-$j$, column-$\ell$ entry is

$$ (\partial_u^{[\ell]}\Delta^jF)(z,u_i(z)). $$

The two operators commute. Equations (2.5)–(2.6) imply that this entry has vanishing order at least $\max(0,L_\ell-j)$. To justify the truncation, one differentiation of a polynomial along $u_i$ differs from its $\Delta$ derivative by a multiple of $(z-\xi_i)^{p-1}$. After $j$ differentiations the cumulative error starts at order at least $p-j$. Since $L_\ell\le p$, it cannot affect the claimed order $L_\ell-j$. This also covers $L_\ell=p$.

The capacities $L_\ell$ are nonincreasing. For $a\ge b$ and $x\ge y$,

$$ (a-x)_++(b-y)_+\le(a-y)_++(b-x)_+. $$

Exchanging out-of-order pairs shows that the minimum determinant-term order pairs $L_0,L_1,\ldots,L_e$ with row indices $e,e-1,\ldots,0$. Therefore

$$ \operatorname{ord}_{\xi_i}W_F\ge \sum_{\ell=0}^e\max(0,L_\ell-e+\ell)=\mu_e. \tag{2.10} $$

The $n$ residues are distinct, so (2.9)–(2.10) require $n\mu_e\le B_e$, contrary to the hypothesis for the actual degree $e$. The reduced scaled matrix has full row rank, and so does the original matrix. This proves Theorem 1.

When $J\le p$ and $e<J$, the multiplicity is simply $J-e$. The mixed formula retains the extra conditions from $J>p$; replacing it by a first-channel count would discard those conditions. Checking only $e=d$ is not part of the general theorem.

2.5 Colliding first residues and unequal capacities

The retained Fermat quotient also allows first residues to collide. Let the nodes be distinct modulo $p^2$, and allow node $i$ to have a capacity $1\le J_i\le R$. Group nodes by their first residue $\xi_i\in k$. Within each group their local series have the form

$$ u_i(z)=c_i+v_\xi(z),\qquad v_\xi(\xi)=0, $$

since every nonconstant coefficient in (2.2) depends only on $\xi$. The constants $c_i$ in a group are distinct. Indeed, if $\lambda_i=\lambda_0+p\eta$, then

$$ \frac{\lambda_i^p-\overline\lambda_i}{p} \equiv\frac{\lambda_0^p-\overline\lambda_0}{p}-\overline\eta \pmod p. \tag{2.11} $$

Thus equality of both coordinates is equivalent to equality modulo $p^2$. The coordinate interpretation is classical [10]; we use it in the following extension of the coefficient-determinant proof.

For each actual degree $e\le d$ and first residue $\xi$, form the multiset

$$ \left\{L_{i,\ell}:=\min(p,\max(0,J_i-\ell p)): \xi_i=\xi,\ 0\le\ell\le e\right\}. $$

Let $a_{\xi,0}\ge\cdots\ge a_{\xi,e}$ be its largest $e+1$ members. Set

$$ \mu_{e,\xi}=\sum_{k=0}^e\max(0,a_{\xi,k}-e+k). \tag{2.12} $$

Corollary 4 (grouped mixed jets). Under $K<p<R\le p^2$, inertness of $p$, and distinctness modulo $p^2$, the matrix with capacities $J_i$ has full row rank if

$$ \sum_\xi\mu_{e,\xi}>B_e\qquad(0\le e\le d). \tag{2.13} $$

To prove it, choose the largest-capacity columns as prefixes of Hasse orders at each center $c_i$. This is possible because $L_{i,\ell}$ decreases with $\ell$; break ties in favor of lower orders. Let the chosen lengths be $t_i$, with $\sum t_i=e+1$. Evaluation of polynomials of $u$ degree at most $e$ in these Hasse prefixes has confluent Vandermonde determinant

$$ \prod_{i<j}(c_j-c_i)^{t_it_j}, $$

up to sign, a nonzero constant. Translation by $v_\xi(z)$ has determinant one, so this change of columns preserves the valuation of $W_F$ at $\xi$. The entry of row $j$ in the column belonging to $(i,\ell)$ has order at least $\max(0,L_{i,\ell}-j)$ by the same local chain rule. The assignment inequality in Section 2.4 yields (2.12). Summing over distinct first residues contradicts $\deg W_F\le B_e$, proving the corollary.

If second coordinates repeat, retaining one representative with the largest capacity is safe; this proof claims no additional information from further digits. When all first residues are distinct, (2.13) is the unequal-capacity form of Theorem 1. Large residue groups can saturate because only $e+1$ evaluation columns enter the coefficient determinant. The extension does not by itself establish a smaller asymptotic prime scale for the elongated quadratic-node family or a new discriminant endpoint.

3. An explicit criterion approaching the dimension count

We prove (1.3). Set $Q=\lfloor J/p\rfloor$. If $e<Q$, every capacity $L_0,\ldots,L_e$ equals $p$, and

$$ \mu_e=(e+1)(p-e/2),\qquad B_e\le(e+1)pR/K. $$

Since $e<p$ and $n\ge2R/K$, we obtain $n\mu_e>B_e$. If $e\ge Q$, all nonzero capacities occur among $L_0,\ldots,L_e$, whence

$$ \mu_e\ge J-(Q+1)e\ge J-\frac{RJ}{p^2}-\frac Rp. \tag{3.1} $$

Also $D_m\le p(R-mp)/K$. Writing $R=dp+s_0$, $0<s_0\le p$, gives

$$ 2p\sum_{m=0}^d(R-mp)=R^2+pR+s_0(p-s_0)<(R+p)^2. $$

It follows that $B_e\le\sum_{m=0}^dD_m<(R+p)^2/(2K)$. Condition (1.3) now proves every degree inequality of Theorem 1.

When the bracket is positive, a sufficient integer node count is

$$ n=\max\left(2S,1+\left\lfloor \frac{(R+p)^2}{2K[J-RJ/p^2-R/p]} \right\rfloor\right). \tag{3.2} $$

Suppose $K$ is fixed and $J/R\to\epsilon\in(0,1/4)$, $p/R\to0$, $R/p^2\to0$. The $2S$ alternative is eventually inactive, since the other term divided by $S$ tends to $1/(2\epsilon)>2$. Using $M=R(R/K+1)/2$ gives

$$ \frac{nJ}{M}=1+O\left(\frac pR+\frac R{p^2}+\frac1p+\frac1R\right). \tag{3.3} $$

This is a conditional rank consequence; it does not by itself supply inert primes, distinct residues or a discriminant bound. At $p\asymp R^{3/4}$ its overhead is $O(R^{-1/4})$; the next section makes that scale available explicitly for the quadratic nodes. If distinct residues are available at $p\asymp R^{2/3}$, (3.3) instead gives $O(R^{-1/3})$. That smaller-prime geometry is not assumed in our application.

4. The explicit nearest-node family

4.1 Inertness and a prime below the norm threshold

Use the parameters (1.4). Bertrand's postulate [9] supplies $A_0<p<2A_0$. The strict upper inequality follows because $2A_0$ is even and greater than two. For $D\ge10000$,

$$ \frac{2A_0}{D}\le\frac{11}{5D^{1/4}}+\frac2D \le0.2202<\frac{1-3/10000}{4}=0.249925. \tag{4.1} $$

Thus $p<(D-3)/4$. Under $h(-D)=1$, a non-inert prime has a principal integral ideal of norm $p$. Its generator is nonrational, since the norm of a rational integer is a square. Writing a nonrational algebraic integer as $x+y\sqrt{-D}/2$, with nonzero integer $y$ and the appropriate integer or half-integer $x$, shows its norm is at least $D/4$. Hence $p$ is inert. This elementary argument uses no discriminant enumeration.

4.2 Geometry without circular dependence

Let $N=\lceil3D/2\rceil$. The standard quadratic-lattice covering estimate [1], also used in [3, Section 4.2], gives for the nearest nodes

$$ |\lambda_i|\le\sqrt{\frac{i\sqrt D}{2\pi}}+\frac{\sqrt{D+4}}4. \tag{4.2} $$

The diameter of the containing disk for the first $N$ nodes therefore satisfies

$$ \frac{2\rho_N}{D^{3/4}} \le\sqrt{\frac{3+2/D}{\pi}} +\frac{\sqrt{1+4/D}}{2D^{1/4}} \le1.027247596<1.1. \tag{4.3} $$

Both terms decrease with $D$; the numerical bound is directed at $10000$. It remains to prove the specified $n$ is at most $N$ before using this geometry for its nodes.

The floor bounds give

$$ \frac{29D}{4}-\frac{259}{4}<R\le\frac{29D}{4},\qquad J>\frac{377D}{800}-\frac{4167}{800}. $$

Consequently

$$ \frac HD\ge h_-(D):= \left(\frac{377}{800}-\frac{4167}{800D}\right) \left(1-\frac{725}{121\sqrt D}\right) -\frac{145}{22D^{3/4}}, \tag{4.4} $$

where $h_-$ is positive and increasing for $D\ge10000$. Moreover,

$$ \frac{(R+2A_0)^2}{86D^2}\le b_+(D):= \frac1{86}\left(\frac{29}{4}+\frac{11}{5D^{1/4}}+\frac2D\right)^2, \tag{4.5} $$

which decreases. Directed evaluation proves

$$ \frac32h_-(10000)-b_+(10000)>0.0050176383940755. \tag{4.6} $$

Thus $NH>(R+2A_0)^2/86$. Since $N$ is integral, the second term in the maximum defining $n$ is at most $N$. The first also is, because $2S\le29D/86<3D/2$. This proves $n\le N$ without presupposing the node geometry.

By (4.3), $2\rho_n\le2\rho_N<A_0<p$. A nonzero multiple of $p$ in $\mathcal O$ has modulus at least $p$, since the norm of a nonzero algebraic integer is a positive integer. The chosen nodes consequently have distinct residues modulo $p$.

Finally, $K<p<R$ and $R/p^2<725/(121\sqrt D)<1$. The bracket on the left of (1.3) exceeds $H$ and its right side is below $(R+2A_0)^2/86$. Definition (1.4) implies (1.3), proving Theorem 2. Its asymptotic follows directly from $A_0\asymp D^{3/4}$, $R\asymp D$ and (1.4). All finite thresholds used in the application are the explicit ones proved above.

The explicit divided-Frobenius column budget approaches the necessary ratio one as the discriminant parameter increases, while the earlier full-triangle Wronskian sufficient ratio stays near 1.521. These are theorem parameter budgets, not measured matrix ranks.

Figure 1. Exact parameter budgets at 25 displayed integer values of $D$. The earlier criterion [2] uses $n_W=\max(S,1+\lfloor(M-S(S+1)/2)/(J-S+1)\rfloor)$. For this family its column ratio tends to $546/359$, while Theorem 2's ratio tends to one. This compares sufficient rank conditions, without asserting the existence of fields at the plotted parameters or comparing their complete determinant costs. The figure data are exact rationals; the curves between points are visual guides.

5. The imported determinant comparison

We state the precise local inputs used in Theorem 3. Their proofs are in [3, Sections 2.1–2.4] and [4, Sections 3–4]; the Frobenius tower is credited to [1, Section 4.3]. Theorem 1 replaces their independence input. It does not replace their arithmetic or metric estimates.

5.1 Integral Gram and row divisibility

For the original Hasse matrix $A$, multiply order-$j$ columns by

$$ C_j=\prod_{q\ \mathrm{prime}}q^{\lfloor j/q\rfloor} $$

and call the resulting integral matrix $A_C$. These factors are distinct from the auxiliary-prime scalings used only in Section 2. Full row rank makes

$$ \Gamma=\det(A_CA_C^*)=\sum_{|E|=M}|\det(A_C)_E|^2 $$

a positive rational integer. Here each squared modulus is the norm of an algebraic integer.

For each prime $q$, let $a_q(r,s)$ be the largest $a$ for which $(r-qt,s+t)\in\Lambda$ for $0\le t\le a$. Explicitly,

$$ a_q(r,s)= \begin{cases} \lfloor r/q\rfloor,&q\ge K,\\ \min\left(\lfloor r/q\rfloor, \left\lfloor\frac{R-1-r-Ks}{K-q}\right\rfloor\right),&q<K. \end{cases} $$

Set

$$ b_q(r,s)=a_q(r,s)+ \left\lfloor\frac{r-qa_q(r,s)}{q^3}\right\rfloor +\sum_{\nu\ge1}\left\lfloor\frac{s}{q^{2\nu}}\right\rfloor. \tag{5.1} $$

The local integral triangular basis of [3, (2.1)–(2.4)] gives

$$ \log\Gamma\ge2\sum_{(r,s)\in\Lambda} \sum_{q\ \mathrm{inert}}b_q(r,s)\log q. \tag{5.2} $$

All rows participate, so no unknown greedy subset or rank optimization remains. At non-inert primes exceeding $K,S$ and with $q^2>R$, only the primary credit $\lfloor r/q\rfloor$ is lost. Under class number one, the principal-norm overcount in [3, (2.8)] gives

$$ \ell_D(r)=\log R \sum_{\substack{t,y\ge1\\ty^2D\le4r}} \left(1+\sqrt{r/t-Dy^2/4}\right). \tag{5.3} $$

For fixed $r$, it decreases with $D$, including the removal of boundary terms. It counts composite norms and repeated representations as well as primes, so it is an upper bound rather than a prime census. Replacing the inert sum in (5.2) by the sum over all primes and subtracting $\ell_D(r)$ is valid whenever the displayed non-inert-prime conditions hold.

5.2 Physical disk-Sobolev upper bound

Let $\rho$ contain the nodes, let $T>1+1/(2\rho)$, and put $\mathcal R=\rho T$. Distinct integral nodes are separated by at least one, so disjoint sampling disks of radius $1/2$ are available. Set

$$ \sigma_Q=\sum_{\substack{q^\ell\le1000\\q\ \mathrm{prime},\ \ell\ge2}} \frac{\log q}{q^\ell},\qquad H_Q=\sum_{\substack{q^\ell\le1000\\q\ \mathrm{prime},\ \ell\ge2}}\log q. $$

Legendre's formula implies $C_j/j!\le e^{H_Q-\sigma_Qj}$. The monic continuous disk basis and weighted sampling theorem [4, (3.7)–(3.9)], scaled to physical monomials as in [3, (2.5)], give

$$ \log\Gamma\le P+\sum_{(r,s)\in\Lambda}\log h_{r,s}, \tag{5.4} $$

where

$$ \begin{aligned} h_{r,s}&=4S^2e^{2H_Q}\kappa\mathcal R^{2r+2s+2} \widetilde h_{r,s}^{\alpha} \sum_{j=0}^{\min(J-1,r)}e^{-a_j} \left(\frac{e^{-\sigma_Q}}{\mathcal R}\right)^{2j} (r)_{\underline j}(s+\alpha+1)_{\overline j},\\ \widetilde h_{r,s}^{\alpha} &=\frac{r!s!\Gamma(r+\alpha+1)\Gamma(s+\alpha+1)} {(r+s+\alpha+1)\Gamma(r+s+\alpha+1)^2},\\ \kappa&=\left(\frac{T^2-1}{T^2-(1+1/(2\rho))^2}\right)^{\alpha+J-1},\\ P&=\sum_i\sum_{0\le j<J} \left[a_j-(\alpha+j)\log\left(1-\frac{|\lambda_i|^2}{\mathcal R^2}\right)\right]_+. \end{aligned} \tag{5.5} $$

The subscripts under and over the factorials denote falling and rising products. The real numbers $a_j$ are free column prices. Their entire positive restoration is paid in $P$. The factor $4S^2$, area Jacobian $\mathcal R^2$, and weight variation $\kappa$ are retained.

Thus a negative upper bound for

$$ P+\sum_{(r,s)\in\Lambda} \left(\log h_{r,s}-2\sum_qb_q(r,s)\log q+2\ell_D(r)\right) \tag{5.6} $$

contradicts (5.2)–(5.4). All reported gaps are the negative of this upper bound divided by $W=R^2S/3$.

6. Eight closed discriminant intervals

For a closed interval $[D_-,D_+]$, fix

$$ K=43,\quad S=\left\lfloor\frac{29(D_--3)}{172}\right\rfloor, \quad R=43S,\quad J=\left\lfloor\frac{13R}{200}\right\rfloor, \quad\alpha=\lfloor S/2\rfloor,\quad T=129/128. \tag{6.1} $$

The following complete comparisons use a $1024\times1024$ row grid and one retained list of 65 rational prices, stored in results/divided_frobenius_prices_10000_20261027_witness.json. The price is interpolated affinely at the equally spaced derivative knots; its physical value at order $j$ is $a_j=Ra(j/R)$. The exact witness and all rounding conventions are supplied with the certifier.

| Closed interval | $S$ | $J$ | $n$ | Auxiliary prime or interval | Gap exceeds | |---|---:|---:|---:|---|---:| | $[9500,9750]$ | 1601 | 4474 | 12883 | $2239$ | 0.0033927302426302 | | $[9750,10000]$ | 1643 | 4592 | 13215 | $2297$ | 0.0139149713381291 | | $[10000,10200]$ | 1685 | 4709 | 13911 | $(4709,5009]$ | 0.0087710473170558 | | $[10200,10600]$ | 1719 | 4804 | 14186 | $(4804,5104]$ | 0.0122903598572444 | | $[10600,11200]$ | 1786 | 4991 | 14736 | $(4991,5291]$ | 0.0232662823585649 | | $[11200,12100]$ | 1887 | 5274 | 15563 | $(5274,5574]$ | 0.0390086575969579 | | $[12100,14000]$ | 2039 | 5699 | 16359 | $2851$ | 0.0689486357331097 | | $[14000,18000]$ | 2359 | 6593 | 18903 | $3301$ | 0.0980192433362680 |

6.1 Inert auxiliary primes and full rank

The four fixed primes are checked by exact trial division and lie below $(D_--3)/4$. Section 4.1 proves their inertness for the entire interval. For a candidate interval $(A,B]$, the certifier lists all primes by exact trial division and bounds its number of non-inert members as follows.

Fix the actual $D\ge D_-$ and put $Y=\lfloor\sqrt{4B/D}\rfloor$. If $Y=0$, every candidate is inert. Otherwise, a non-inert prime is a norm $x^2+Dy^2/4$ with $1\le y\le Y$. At each fixed $y$, nonnegative $x$ lies on one grid of spacing one. For $y<Y$, the length of its allowed interval is at most $(B-A)/(2\sqrt{A-By^2/Y^2})$, provided the denominator is positive. For $y=Y$ its length is at most $\sqrt{B-A}$. Therefore the number of non-inert primes is at most

$$ Y+\sqrt{B-A}+ \sum_{y=1}^{Y-1}\frac{B-A}{2\sqrt{A-By^2/Y^2}}. \tag{6.2} $$

This uses $D\le4B/Y^2$ in the lower endpoint. It overcounts representations and composite norms. The certifier checks every $1\le Y\le\lfloor\sqrt{4B/D_-}\rfloor$, all denominators, and that (6.2) is strictly below the exact candidate-prime count. At least one candidate is consequently inert; a particular prime is not assumed inert uniformly in $D$.

For each fixed prime, the program checks every inequality $n\mu_e>B_e$ of Theorem 1. For each candidate interval, $J\le p$ and the node count satisfies the special-case inequality for every candidate; all smaller actual-degree inequalities follow or are checked. This pays the largest required node count over the candidate set.

6.2 A single geometry for the entire interval

Put

$$ b_+=\sqrt{\frac{n\sqrt{D_+}}{2\pi}},\qquad \delta_+=\frac{\sqrt{D_++4}}4, \qquad\eta_+=\delta_+/b_+. $$

Choose the rational $\rho$ by rounding $b_++\delta_+$ upward to a multiple of $2^{-40}$. For every $D\in[D_-,D_+]$, (4.2) implies

$$ |\lambda_i|\le\rho,\qquad \left|\frac{\lambda_i}{\rho}\right|^2 \le q_{\eta_+}(i/n),\qquad q_\eta(v)=\left(\frac{\sqrt v+\eta}{1+\eta}\right)^2. \tag{6.3} $$

The diameter upper bounds for the eight intervals respectively are below $950,968,997,1016,1050,1099,1170,1338$. Each is below the smallest candidate prime, so all node residues are distinct at the selected inert prime. Theorem 1 supplies full rank throughout each interval.

This same $\rho$ enters every term of (5.5), including the derivative scale $L=\log(R/\rho)-\sigma_Q$, normalized sampling radius $1/(2\rho)$, and restored price profile. There is no change of physical row normalization between rank and norm estimates.

6.3 Directed costs and interval coverage

For each row-cell the certifier uses exact integer cell counts and the upper disk-Sobolev phase from [4, Sections 3–4]. The arithmetic credits (5.1) use exact integer floors, with $\log q$ rounded downward. Formula (5.3) is evaluated at $D_-$ and the cell's upper $r$ endpoint. In all intervals,

$$ \frac{D_--3}{4}>\max(K,S),\qquad \left(\frac{D_--3}{4}\right)^2>R, $$

so every non-inert prime satisfies the hypotheses of Section 5.1. The restored price uses (6.3), the increasing radial integrand, its exact primitive from [4, (3.5)], and the endpoint correction for the node sum. Convex chords account for all integer derivative orders.

All transcendental calculations use 256-bit Arb enclosures. Row tables are rounded outward to exact multiples of $2^{-40}$ and then summed with integer counts. The floating optimizer that proposed the finite-interval price witness reached its iteration cap. Its convergence and optimality are not proof premises: the retained rational prices are verified directly.

The resulting exact dyadic gaps are positive and imply the table. Each statement excludes the entire closed interval, not merely sampled endpoints. The intervals meet and cover $[9500,18000]$. To finish Theorem 3, we supply this method's own uniform tail next.

7. A uniform full-triangle tail for every $D\ge18000$

Set $D_0=18000$, $X=(D-3)/4$, $X_0=17997/4$ and

$$ K=43,\quad S=\lfloor29X/K\rfloor,\quad R=KS,\quad J=\lfloor13R/200\rfloor,\quad\alpha=\lfloor S/2\rfloor, \quad n=\lceil3D/2\rceil,\quad T=129/128. \tag{7.1} $$

Theorem 2 gives full rank at an initial subset of these nodes. Adding the remaining columns preserves it. At the anchor, $S_{\min}=3034$, $R_{\min}=130462$, $J=8480$, $n=27000$, and $M=197976085$.

7.1 Uniform geometry and all physical costs

The integer node rule satisfies

$$ c_-^n:=\frac6{29}\le\frac nR\le c_+^n:=\frac6{29}+\frac{6K/29+11/2}{KS_{\min}} =\frac{1566379}{7566796}. \tag{7.2} $$

Indeed, $R\le29(D-3)/4$ gives the lower bound, while $D<4K(S+1)/29+3$ and $n<3D/2+1$ give the upper bound. Define

$$ \begin{aligned} R_*&=29X_0-K,& \nu_-&=c_-^n(29-K/X_0),&\nu_+&=29c_+^n,\\ \eta_0&=\frac14\sqrt{\frac{2\pi(D_0+4)}{\nu_-X_0\sqrt{D_0}}},\\ C_+&=\frac12\log\frac{\nu_+}{\pi} +\frac14\log\frac{D_0}{D_0-3}+\log(1+\eta_0),\\ \rho(X)&=e^{C_+}X^{3/4},&\rho_0&=\rho(X_0). \end{aligned} \tag{7.3} $$

The proof of [3, Section 4.2] uses only these lower and upper node ratios and (4.2). It gives a containing radius $\rho(X)$ and normalized profile $q_{\eta_0}(i/n)$ for every $D\ge D_0$. Numerically, $\eta_0<0.044189817199$ and $C_+<0.367064361373$. The derivative scale obeys

$$ L(D)\le L_0+\frac14\log t,\qquad L_0=\log29+\frac14\log X_0-C_+-\sigma_Q, \qquad t=\frac{D-3}{17997}. \tag{7.4} $$

The same physical radius is used in this scale and in (5.5). The certifier checks $T>1+1/(2\rho_0)$; the sampling ratio decreases thereafter.

7.2 Pointwise arithmetic and row cost tables

We use the explicit all-prime bound from [4],

$$ \Sigma_2=\sum_{q\le100000}\frac{\log q}{q(q-1)} +\frac{\log100000+1}{99999} \ge\sum_q\frac{\log q}{q(q-1)}, $$

where $q$ ranges over primes. Legendre's formula and $\log(r!)\ge r\log r-r$ give the primary lower credit. The small-prime residual and tower corrections are (5.1). Their uniform cell enclosures are precisely [3, (4.5)], whose parameters $K,S_{\min},R_{\min}$ are unchanged. The rowwise norm loss is [3, (4.6)]: writing $A_*=29/4$ and $\omega_R=A_*-(87/4+K)/D_0$, twice (5.3) divided by $R$ is at most

$$ \frac{2\log(A_*D)}{\omega_RD} \#\{(t',y):t'y^2\le4A_*x_h\} +\frac{2\log(A_*D)}{\omega_R\sqrt D} \sum_{t'y^2\le4A_*x_h}\sqrt{A_*x_h/t'-y^2/4}. \tag{7.5} $$

Both prefactors decrease on $D\ge D_0$, so freeze them at the anchor. No asymptotic prime estimate or finite discriminant data enters this step.

Use a grid of size $N_g=640$. Cell $(i,j)$ has degree intervals $[\lfloor iR/N_g\rfloor,\lfloor(i+1)R/N_g\rfloor)$ and the analogous $s$ interval. Put

$$ \begin{aligned} x_l&=\max(0,i/N_g-1/R_{\min}),&x_h&=(i+1)/N_g,\\ y_l&=\max(0,j/(KN_g)-1/R_{\min}),&y_h&=(j+1)/(KN_g),\\ \beta_+&=1/(2K),&\beta_-&=\beta_+-1/(2R_{\min}),\\ \lambda&=\lceil(13/200)R_{\min}\rceil/R_{\min}. \end{aligned} $$

These enclose every actual $r/R,s/R,\alpha/R,J/R$. The 65 rational prices in results/sobolev_jacobi_tail_18000_20261027_witness.json are the published witness of [4]. They need not be optimal for the present nodes. Their row maxima and restored spatial charges are recomputed with (7.3).

For clarity, the finite allowance accompanying the row entropy and derivative phase is

$$ \begin{aligned} \mathcal E_0={}&\frac{2\log(\rho_0T) +\log((1+\beta_+)29X_0+1)+\log((13/200)29X_0+1)}{R_*}\\ &+\frac{\log(4(29X_0/K)^2)+2H_Q}{R_*} +(\beta_++13/200) \log\frac{T^2-1}{T^2-(1+1/(2\rho_0))^2}. \end{aligned} \tag{7.6} $$

This is [3, (4.7)] with the new radius. The first numerator has logarithmic derivative at most $11/2$ and exceeds $11/2$ at the anchor; its ratio to $X$ therefore decreases. The weight term decreases with the sampling ratio. This pays the finite factorial/diagonal remainders, derivative sum, sampling factor, column-factorial allowance and area term. Its directed value is below $0.007440005421$.

Let $C_{ij}$ be the resulting upper normalized anchor cost from [3, (4.8)], recomputed using (7.2)–(7.6). The same pointwise argument gives

$$ \frac{c_{r,s}(D)}R\le C_{ij}+G_{ij}\log t, \qquad G_{ij}=-2x_l+\frac32(x_h+y_h)+\frac\lambda2, \tag{7.7} $$

where $c_{r,s}$ is the parenthesized row cost in (5.6). The three growth terms come respectively from arithmetic, physical radius, and the derivative scale (7.4). All other terms have been frozen by monotonicity. Both tables are rounded upward to multiples of $2^{-40}$. We keep every signed entry of both tables. Individual entries of $G$ may be positive.

7.3 Exact degree uniformity and restored prices

For each residue $v\in\{0,\ldots,N_g-1\}$ write $S=N_gq+v$. The exact cell count is a quadratic polynomial in $q$, including boundary and fringe cells; see [4, Section 4]. Thus the signed weighted sum divided by $S^2$ is of the form

$$ \frac{Aq^2+Bq+C}{(N_gq+v)^2},\qquad q\ge\max\left(2,\left\lceil\frac{S_{\min}-v}{N_g}\right\rceil\right). $$

Its derivative has numerator $(2Av-BN_g)q+Bv-2CN_g$. Its supremum on the real half-line is attained at the endpoint, at the limit, or at its single admissible critical point. All coefficients and comparisons are rational. Checking all 640 residues therefore bounds every integer $S\ge3034$, without a sampled-degree extrapolation. Apply this procedure separately to the complete $C$ and $G$ tables. After multiplication by $3/K$, which is the normalization $R/W$ times $S^2$, the resulting bounds are

$$ U_{\rm rows}=-\frac{113428752617591176611}{483834421621364359168}, \qquad G_{\rm all}=-\frac{161803059240843993}{900719925474099200}. \tag{7.8} $$

The factor $1/2$ that occurs in the stored quadratic coefficients is included by the checker; the mathematical cell counts themselves are integers.

At each price knot $(\xi_k,a_k)$ put

$$ F_k=\int_0^1 [a_k-(\beta_++\xi_k)\log(1-q_{\eta_0}(v)/T^2)]_+\,dv. $$

The radial primitive and convex chord bound of [4] give, with $I_F$ the integral of the knot chords, $E_F=\sum_{k=0}^{63}2(F_k+F_{k+1})$, and $E_{\max}$ the largest nonnegative radial endpoint price,

$$ P_0=3Kc_+^n(I_F+E_F/R_{\min}) +\frac{3K\lambda}{R_{\min}}E_{\max}. \tag{7.9} $$

The final term pays the radial endpoint correction for every available column; $E_F$ pays integer-order rounding. The recomputed value is $P_0<0.112857403872$. In particular, no price charge from an incompatible node rule or radius is reused.

7.4 Contradiction and the certified domain

Combining (5.6)–(7.9), a hypothetical field with $D\ge18000$ would satisfy

$$ 0\le U_0+G_{\rm all}\log\frac{D-3}{17997}, \qquad U_0=U_{\rm rows}+P_0. \tag{7.10} $$

The directed result is

$$ -U_0>0.1215797249241356,\qquad -G_{\rm all}>0. $$

This is impossible for every $D\ge18000$. The algebraic zero of (7.10) falls below $18000$, outside the proved domain, and is not used. The eight closed intervals of Section 6 cover the rest of $[9500,\infty)$ and prove Theorem 3. No published global discriminant theorem is needed to close the upper range.

8. Supporting controls, comparisons and limitations

The proof is supported by exact finite-field and integral controls. They compare the actual scaled Hasse entries with local formulas assembled by a separate code path, rather than testing only a symbolic copy of the claimed formula.

| Control | Coverage | Purpose | |---|---|---| | Two divided layers | 138 cases, 496686 entries | Integral division and the boundary $J=p$ | | Three through six layers | 66 cases, 919006 entries | Higher layers, nonzero lift constants, raw rank loss | | Differential connection | 150 exact identities | $\Delta^p=-\partial_u$ and failure at degree $p$ | | Mixed higher Hasse columns | 102 cases, 1565852 entries | Column scaling, including 667140 entries with order at least $p$ | | Short final channels | 54 mixed cases | Capacities no larger than the coefficient degree | | Multiplicity assignments | 203 exhaustive assignments | The minimum determinant-term order | | Colliding first residues | 156 cases, 6590934 entries | Second coordinates, unequal capacities and saturated groups | | Sufficient criterion | 1019 cases, 22068 actual-degree tests | (1.3) implies every inequality of Theorem 1 | | Uniform constants | 256-bit directed evaluation | (4.3)–(4.6), followed by the written monotonicity proof |

The collision controls check 4978 second-coordinate identities. In 58 cases the node count exceeds $p^2$, so distinct first residues are impossible; in 84 cases retaining only one node per first residue loses rank. For example, at $p=3,K=2,S=4,J=3$, ten nodes in five first-residue groups have rank 20, while one node from each group has rank 15. The strict criterion margin is one. These are exact constructed quadratic-ring cases, not claims about hypothetical discriminants.

The controls do not replace the proofs and are not independent scientific review. Ten exact parameter examples through $D=10^{12}$ illustrate Theorem 2; they do not establish its infinite range.

At the fixed comparison resources of [3], $K=43,S=3034,J=8480,n=28138$, the earlier complementary-rank certificate gave gap $>0.0067773604623792$; its improved total-degree/prefix comparison gave $>0.0072546834310851$. The original $J\le p$ divided-layer theorem certifies all $197976085$ rows at these resources and gives $>0.1201502773842025$, retaining the same physical radius and prices. This is a matched fixed comparison, not the uniform proof of Section 7. It does not assert that the earlier sufficient criteria described the actual rank sharply.

A second comparison isolates the higher-Hasse extension. On the complete interval $[10000,10200]$, the fixed prime $2357<(10000-3)/4$ is inert under the hypothesis. The mixed criterion requires $13550$ nodes, instead of $13911$ for the original prime-interval criterion. Its node diameter is below $984$, and a recomputation at the same rational prices and grid gives gap $>0.0251845344681676$, compared with $>0.0087710473170558$ in Section 6. The stronger single-interval result is supplementary; the complete bridge uses the eight displayed certificates.

Theorem 1 requires an inert prime with distinct node residues and $R\le p^2$. Section 2 explains the cyclicity obstruction beyond the latter range. The near-dimension statement concerns a sufficient rank criterion, not the exact rank threshold of every matrix. The discriminant application retains published local divisibility and Sobolev inputs. It does not reprove their full frameworks, classify the fields below $9500$, or regenerate an external census. The separately qualified global consequence in Section 8.1 also imports [1]’s reviewed cutoff.

Corollary 4 also retains colliding first residues and unequal capacities. Its individual truncated channels cannot be replaced by their total. The application in Sections 4–7 uses distinct first residues and equal capacities, so it does not depend on this extension.

8.1 A matched application with the published Meixner estimate

The same rank theorem also improves the node budget in the published Meixner/composition/Schur comparison [1, Section 6.4(4); 11, Sections 2–5.5]. This is a separate application from the complete proof in Sections 5–7. It concerns fundamental discriminants with $h(-D)=1$ and $D\equiv3\pmod4$ in the closed interval $[499,503]$. The published discrete norm-product subtraction uses this congruence; we do not assert a comparison for every real number in the interval.

Fix

$$ \begin{aligned} P_{\max}&=312000000,&K&=43,&S&=8706976,\\ R&=374399968,&J&=46799996,&M&=1629945955088368,\\ n&=35394400. \end{aligned} \tag{8.1} $$

There are $1649$ primes in $(150000,170000]$. At any fixed discriminant in our interval, a non-inert prime here is unramified and split, hence under $h=1$ is the norm of an algebraic integer. After conjugation and negation, write that norm as $x^2+Dy^2/4$ with $y\ge1$ and $x\ge0$. For fixed $y$, the permissible $x$ lie on a translate of the integer grid. The case $y=0$ would give a square, not a prime. Thus, with $A=150000$ and $B=170000$, the number of non-inert primes is at most

$$ \sum_{y=1}^{\lfloor\sqrt{4B/499}\rfloor} \left[1+\sqrt{B-499y^2/4} -\sqrt{\max(0,A-503y^2/4)}\right] <1453.129359<1649. \tag{8.2} $$

Each actual fixed-$D$ segment is enclosed by the displayed segment, and an interval of length $l$ contains at most $1+l$ points of a unit grid. Distinct primes require distinct norm representations at that fixed $D$; composite norms are overcounted. The directed evaluation of (8.2) therefore forces an inert prime without identifying the hypothetical field.

For every prime in the specified interval, $K<p<R<p^2$. Checking every actual degree in Theorem 1 gives $3867213$ integer inequalities, with smallest margin $n\mu_e-B_e=23702027808$. The nearest-node diameter obeys

$$ 2\rho_n\le 2\sqrt{\frac{n\sqrt{503}}{2\pi}}+\frac{\sqrt{507}}2 <22491.416<150000. \tag{8.3} $$

Consequently the nodes have distinct residues at the inert prime and the original integral matrix has full row rank. None of the rank scalings is used to alter its metric or valuation costs.

We now apply the local comparison in [1, Section 6.4(4)]. It uses the spectral parameters $(A_{\rm MS},v,\beta,a_0)=(1/5,7/10,-4/5,3/50)$, $Q_{\rm guar}=J$, two- and three-factor composite subtraction, and the published $128$ radial, $256$ column and $512$ occupancy cells. The four-factor branch is disabled. The normalized arithmetic lower bound $L$ is compared with

$$ U=\frac{W_1+W_2}{W}\log\rho_{n,\rm hi} +U_{\rm MS}+\frac{M c_j+2M\log M}{W},\qquad W=R^2S/3, \tag{8.4} $$

where $W_1=\sum_\Lambda2r$, $W_2=\sum_\Lambda2s$, $c_j$ is the published column prime-power remainder, and $U_{\rm MS}$ is the finite Meixner/composition/Schur upper bound. The physical radius, column remainder and positive counting allowance are retained. A positive $L-U$ contradicts the nonzero minor supplied by Theorem 1.

The author adapter replaces only the original rank/node-choice block of certify_hybrid_k3_interval and adds result capture. It verifies the hashes of all three imported source files before execution; the exact function diff is supplied. The source flag use_wronskian also selects two existing high-occupancy guards. Their numerical conditions hold independently here: $P_{\max}/125=2496000<125^4$ and $J>\lfloor M/n\rfloor$. Thus this flag does not import an unproved Wronskian rank claim at the new node count. Every node-dependent quantity is recomputed, including occupancy, radius, spectral slopes, small-prime capacities, split-prime column floors and column remainders. The relevant caches contain $n$, and the candidate norm range includes $J-1$ and $\lfloor M/n\rfloor$. The source manifest and application note give the complete arguments and audit.

For a matched control, retain the published rank/node-choice block, which gives $n=41793600$, and make no other parameter change. The source's decimal interval implementation gives the following outward-rounded values.

| Quantity | Mixed rank | Published Wronskian rule | |---|---:|---:| | Nodes $n$ | $35394400$ | $41793600$ | | Occupancy $M/(nJ)$, approximate | $0.98399487$ | $0.83333113$ | | Arithmetic lower $L$ | $9.9949384461076672$ | $9.9915378781239405$ | | Common metric upper $U-U_{\rm MS}$ | $9.6200286379874897$ | $9.7114321931986488$ | | Variable metric upper $U_{\rm MS}$ | $0.3237112105144662$ | $0.4035380283714751$ | | Complete gap enclosure | $[0.0511985976057114,0.0512005948607304]$ | $[-0.1234323434461833,-0.1234303461911644]$ |

The binary interval implementation also gives a positive lower gap, $0.0511971293819613$. These are two implementations supplied in one published codebase, retaining its directed binary64 inner calculations. They are not an independent implementation or a new evaluation of the enormous determinant. The change of sign isolates the effect of the rank-certified node budget at the stated support, derivative budget, arithmetic cutoff and spectral prices.

Corollary 5 (with an imported global bound). Combining the exclusion of $D=499$ above with the reviewed implication $h(-D)=1\Rightarrow D\le499$ in [1, Theorem 1.2] gives

$$ h(-D)=1\quad\Longrightarrow\quad D\le498, \qquad\log498=6.21060007702465292428\ldots. \tag{8.5} $$

This corollary explicitly imports a global theorem. It is distinct from Theorem 3, whose complete $D<9500$ route imports only local determinant and metric inputs. No new classification or regeneration of a census follows from this comparison, and no further finite elimination is performed here.

9. Reproducibility and research disclosure

The supplement includes proof notes, author code, pinned dependencies, rational price witnesses, expected exact results, and hashes. Section 8.1 also uses three public source files from [1], fetched by a supplied script from version-specific URLs and checked against pinned SHA256 hashes. These files are accessible without credentials and are not redistributed in the ZIP. From its root, install requirements.txt and run

python code/replay_complementary_rank.py --workers 5

The original 33 sequential stages use at most five CPU workers and passed a fresh directory replay in 293.520 seconds. Five additional stages fetch the pinned public source, regenerate (8.1)–(8.3), run both matched comparisons and check (8.4)–(8.5). The final replay receipt records their combined execution: all 38 stages passed in 400.473 seconds, including the public-source download and hash verification. It recomputed the exact algebra controls, finite interval gaps, full-triangle uniform tail and interval coverage, alongside the earlier comparison controls. REPRODUCTION.json records the measured runtime and each stage's source hash. Stage logs and the larger disposable histogram are regenerated; the histogram is removed after verification. No enormous interpolation matrix is constructed.

code/check_divided_frobenius_complete.py checks the eight exact positive gaps, the tail's proved starting point and growth, and the endpoint logarithm. The rational residue-count coefficients establish the uniform integer-degree coverage. The optional floating price-discovery script has separate dependencies and is unnecessary for the proof replay. Its iteration limit and the retained rational witness are disclosed above.

The public packet also preserves historical comparisons and the earlier complementary-rank manuscript as background. Its README identifies the proof-reading order and the scopes of those earlier claims. The current complete $9500$ argument is Sections 2–7 of this manuscript. That route does not use the global $D<12100$ theorem that appeared in an earlier four-interval corollary.

Mira Cairn is an AI researcher using OpenAI Codex. The proofs, code and reported controls were produced in this research session; there were no subagents or delegated model calls. Author checking, replay of author code, independent implementation, mathematical peer review and formal verification are distinct evidence levels. The new results are submitted for mathematical review. Software and generated data are MIT licensed; the manuscript, proof notes and figures are licensed CC BY 4.0. External mathematical sources are cited and are not redistributed as source code.

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